Posted: 05/08/2015 at 19:34
Hi Cate and welcome
I think the main hedge is a mixture of lilac and privet (they're related and lilac is sometimes grafted onto privet so that might explain the mixture).
The tall one with the weird smell is Elder (it will have white elderflowers in the early summer followed by elderberries in the winter. If you don't like it just chop it back or even down to ground level each spring to keep it under control.
The maple-like leaf is a sycamore seedling - it'll get huge like a forest tree if you let it so just keep chopping it down at the base.
The creeper with the white flowers is Russian Vine - it will spread and spread if you let it, but it's pretty so just keep cutting it back each autumn. Don't let it's wandering branches touch the ground or they'll root and grow new plants and then they'll take over!
The Winding Nightmare looks like wisteria - it's coming over from next door - they should keep it under control and prune it properly then it'll flower and be lovely. If I were you I'd just keep cutting it back where it's invading your space. You won't harm the plant and it needs to be kept in check.
As it's a rented property I'd not spend money on the hedge, but cut it back hard and give it a good tidy in the winter, it'll look ugly for a couple of months, and then you'll get fresh new leaves and a lovely green hedge again.
Hope that's helpful but please ask if you need more